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Linear Systems

The definition of a linear equation given in Chapter 1 can be extended to more variables; any equation of the form

1 1 2 2 n na x a x a x b

for real numbers a1, a2, …, an (not all of which are 0) and b, is a linear equation or a first-degree equation in n unknowns.

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Linear Systems

A set of equations is called a system of equations. The solutions of a system of equations must satisfy every equation in the system. If all the equations in a system are linear, the system is a system of linear equations, or a linear system.

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Linear Systems

The possible graphs of alinear system in two unknowns are as follows.1. The graphs intersect at exactly one point, which gives the (single) ordered pair solution of the system. The system is consistent and the equations are independent.

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Linear Systems

2. The graphs are parallel lines, so there is no solution and the solution set is ø. The system is inconsistent and the equations are independent.

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Linear Systems

3. The graphs are the same line, and there is an infinite number of solutions. The system is consistent and the equations are dependent.

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Substitution Method

In a system of two equations with two variables, the substitution method involves using one equation to find an expression for one variable in terms of the other, and then substituting into the other equation of the system.

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Example 1 SOLVING A SYSTEM BY SUBSTITUTION

Solve the system.

3 2 11x y 3x y

(1)

(2)

Solution

Begin by solving one of the equations for one of the variables. We solve equation (2) for y.

3x y (2)

3y x Add x.

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Example 1 SOLVING A SYSTEM BY SUBSTITUTION

Now replace y with x + 3 in equation (1), and solve for x.

3 2 11x y (1)

3 2( 113)xx Let y = x + 3 in (1).

Note the careful use of

parentheses.

3 2 6 11x x Distributive property

5 6 11x Combine terms.

5 5x Subtract.

1x

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Replace x with 1 in equation (3) to obtain y = 1 + 3 = 4. The solution of the system is the ordered pair (1, 4). Check this solution in both equations (1) and (2).

Example 1 SOLVING A SYSTEM BY SUBSTITUTION

3 2 11x y (1)

Check:

3yx (2)

3( ) 2( 11 4) 1 ?

11 11 True

41 3 ?

3 3 True

Both check; the solution set is {(1, 4)}.

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Elimination Method

Another way to solve a system of two equations, called the elimination method, uses multiplication and addition to eliminate a variable from one equation. To eliminate a variable, the coefficients of that variable in the two equations must be additive inverses. To achieve this, we use properties of algebra to change the system to an equivalent system, one with the same solution set. The three transformations that produce an equivalent system are listed here.

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TransformationS of a Linear System1. Interchange any two equations of the

system.2. Multiply or divide any equation of the

system by a nonzero real number.3. Replace any equation of the system by

the sum of that equation and a multiple of another equation in the system.

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Example 2 SOLVING A SYTEM BY ELIMINATION

Solve the system.

One way to eliminate a variable is to use the second transformation and multiply both sides of equation (2) by – 3, giving the equivalent system

Solution

3 4 1x y (1)

(2)2 3 12x y

3 4 1x y (1)

6 9 36x y Multiply (2) by – 3 (3)

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Example 2 SOLVING A SYTEM BY ELIMINATION

(3)6 9 36x y

Now multiply both sides of equation (1) by 2, and use the third transformation to add the result to equation (3), eliminating x. Solve the result for y.

6 8 2x y Multiply (1) by 2

17 34y Add.

2y Solve for y.

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Example 2 SOLVING A SYTEM BY ELIMINATION

(1)3 4 1x y

Let y = 2 in (1).

Multiply.

Substitute 2 for y in either of the original equations and solve for x.

3 4( ) 12x

3 8 1x

3 9x Add 8.

3x

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Example 2 SOLVING A SYTEM BY ELIMINATION

A check shows that (3, 2) satisfies both equations (1) and (2); the solution set is {(3, 2)}.

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Example 3 SOLVING AN INCONSISTENT SYSTEM

Solve the system.

3 2 4x y

6 4 7x y

(1)

(2)

To eliminate the variable x, multiply both sides of equation (1) by 2.

Solution

6 4 8x y Multiply (1) by 2

6 4 7x y (2)

0 15 False.

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Example 3 SOLVING AN INCONSISTENT SYSTEM

Since 0 = 15 is false, the system is inconsistent and has no solution. As suggested here by the graph, this means that the graphs of the equations of the systemnever intersect. (The lines are parallel.) The solution set is the empty set.

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Example 4SOLVING A SYSTEM WITH INFINITELY MANY SOLUTIONS

Solve the system.

Solution

Divide both sides of equation (1) by 2, and add the result to equation (2).

4 2x y Divide (1) by 2.

8 2 4x y

4 2x y

(1)

(2)

4 2x y (2)

0 0 True.

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Example 4SOLVING A SYSTEM WITH INFINITELY MANY SOLUTIONS

The result, is a true statement, which indicates that the equations of the original system are equivalent. Any ordered pair that satisfies either equation will satisfy the system. From equation (2),

4 2x y (2)

.2 4y x

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Solving An Applied Problem By Writing A System of EquationsStep 1 Read the problem carefully until you understand

what is given and what is to be found.Step 2 Assign variables to represent the unknown values,

using diagrams or tables as needed. Write down what each variable represents.

Step 3 Write a system of equations that relates the unknowns.

Step 4 Solve the system of equations.Step 5 State the answer to the problem. Does it seem

reasonable?Step 6 Check the answer in the words of the original

problem.

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Solving Linear Equations with three Unknowns (Variables)

Some possible intersections of planes representing three equations in three variables are shown here.

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Solving Linear Equations with three Unknowns (Variables)

Some possible intersections of planes representing three equations in three variables are shown here.

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Solving Linear Equations with three Unknowns (Variables)

Some possible intersections of planes representing three equations in three variables are shown here.

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Solving a System

To solve a linear system with three unknowns, first eliminate a variable from any two of the equations. Then eliminate the same variable from a different pair of equations. Eliminate a second variable using the resulting two equations in two variables to get an equation with just one variable whose value you can now determine. Find the values of the remaining variables by substitution. The solution of the system is written as an ordered triple.